Showing papers by "Shivaji Lal Sondhi published in 2003"
TL;DR: It is conjectured that extended critical phases are realized only on bipartite lattices, even in higher dimensions, and that these models have both confined and exponentially deconfined but no critical phases.
Abstract: We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.
185 citations
TL;DR: In this paper, it was shown that the remaining degrees of freedom in the low field plateau live on decoupled kagome planes, and can be mapped to hard core dimers on a honeycomb lattice.
Abstract: The application of a magnetic field along the [111] direction in the spin ice compounds leads to two magnetization plateaux, in the first of which the ground-state entropy is reduced but still remains extensive. We observe that under reasonable assumptions, the remaining degrees of freedom in the low field plateau live on decoupled kagome planes, and can be mapped to hard core dimers on a honeycomb lattice. The resulting two-dimensional state is critical, and we have obtained its residual entropy---in good agreement with recent experiments---the equal time spin correlations as well as a theory for the dynamical spin correlations. Small tilts of the field are predicted to lead a vanishing of the entropy and the termination of the critical phase by a Kasteleyn transition characterized by highly anisotropic scaling. We discuss the thermally excited defects that terminate the plateau at either end, among them an exotic string defect which restores three dimensionality.
118 citations
TL;DR: A modification of Fisher's mapping of two-dimensional Ising models to hardcore dimer models is introduced in which a second dimer model, equivalent to the Ising model, is defined on a generically different lattice derived from the dual.
Abstract: Motivated by recent interest in 2+1 dimensional quantum dimer models, we revisit Fisher's mapping of two-dimensional Ising models to hardcore dimer models. First, we note that the symmetry breaking transition of the ferromagnetic Ising model maps onto a non-symmetry breaking transition in dimer language---instead it becomes a deconfinement transition for test monomers. Next, we introduce a modification of Fisher's mapping in which a second dimer model, also equivalent to the Ising model, is defined on a generically different lattice derived from the dual. In contrast to Fisher's original mapping, this enables us to reformulate frustrated Ising models as dimer models with positive weights and we illustrate this by providing a new solution of the fully frustrated Ising model on the square lattice. Finally, by means of the modified mapping we show that a large class of three-dimensional Ising models are precisely equivalent, in the time continuum limit, to particular quantum dimer models. As Ising models in three dimensions are dual to Ising gauge theories, this further yields an exact map between the latter and the quantum dimer models. The paramagnetic phase in Ising language maps onto a deconfined, topologically ordered phase in the dimer models. Using this set of ideas, we also construct an exactly soluble quantum eight vertex model.
37 citations
TL;DR: The Liouvillian approach to the quantum Hall plateau transition was introduced by Sinova, Meden, and Girvin this article, who formulated it for a general disordered single-particle Hamiltonian.
Abstract: We examine the Liouvillian approach to the quantum Hall plateau transition, as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B 62, 2008 (2000)] and developed by Moore, Zee, and Sinova [Phys. Rev. Lett. 87, 046801 (2001)]. We show that, despite appearances to the contrary, the Liouvillian approach is not specific to the quantum mechanics of particles moving in a single Landau level: we formulate it for a general disordered single-particle Hamiltonian. We next examine the relationship between Liouvillian perturbation theory and conventional calculations of disorder-averaged products of Green functions and show that each term in Liouvillian perturbation theory corresponds to a specific contribution to the two-particle Green function. As a consequence, any Liouvillian approximation scheme may be reexpressed in the language of Green functions. We illustrate these ideas by applying Liouvillian methods (including their extension to ${N}_{L}g1$ Liouvillian flavors) to random matrix ensembles, using numerical calculations for small integer ${N}_{L}$ and an analytical analysis for large ${N}_{L}.$ We find that the behavior at ${N}_{L}g1$ is different in qualitative ways from that at ${N}_{L}=1.$ In particular, the ${N}_{L}=\ensuremath{\infty}$ limit expressed using Green functions generates a pathological approximation, in which two-particle correlation functions fail to factorize correctly at large separations of their energy, and exhibit spurious singularities inside the band of random matrix energy levels. We also consider the large-${N}_{L}$ treatment of the quantum Hall plateau transition, showing that the same undesirable features are present there, too. We suggest that failings of this kind are likely to be generic in Liouvillian approximation schemes.
3 citations